This article, by Professor Gary Rendsburg of Cornell University, provides a number of examples of puns (some of them bilingual) from throughout Tanakh. As you can see from the final page, it is an article within a collection that deals explicitly with this issue.

My personal favourite from the ones that he cites, and he cites many, is a Hebrew/Greek bilingual pun in Proverbs 31:27. Rather than say צופה הליכות ביתה ("she watches over the ways of her household"), a more unusual form of the verb, צופיה, is used, and one which allows the author to pun on the Greek word for wisdom (sofia).

Most importantly, the article is replete with references to other texts (many of them by the same author, though not all of them), which explore other more specific examples of biblical Hebrew wordplay.

For those wondering, the word σοφίης (~sophias) shows up in the Iliad 15:412, which is (roughly, +-200 years) around the time of Shlomo. So it isn't at all unreasonable that he would know the word.
– Double AA♦Feb 5 '13 at 6:21

These are only puns for people who don't pronounce Hebrew properly. The original speakers of those lines as well as the first readers of the text were certainly not in that category.
– Double AA♦Oct 23 '12 at 19:58

4

Puns exist when pronunciations are similar, not only when they’re identical.
– J. C. SalomonOct 23 '12 at 20:19

1

True, but I think our notion of what sounds are similar may be somewhat skewed. ח is probably much closer to ה than to כ.
– Double AA♦Oct 23 '12 at 20:42

Note that ח and כ are classically considered to be pronounced by different parts of the mouth (unlike e.g. ע and א). Anyway, any evidence these were intended as puns?
– msh210♦Oct 24 '12 at 2:31

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Although I think the definition of pun here is being loosely applied. It's more style than pun. I think.
– Seth JFeb 27 '14 at 13:22

I've always been enamored with Bamidbar 19:17: וְנָתַן עָלָיו מַיִם חַיִּים, אֶל-כֶּלִי.
I'm waiting to be at a meal with a person name Kelly, who asks someone (not next to her) to pass the Mayim Chaim (brand of soda pop). When the person passes it over the man sitting next to Kelly, he will place it on his head, reciting this verse.

@DoubleAA Yes, that was it. It also explains why i was confused. I wasn't sure if the pun was צְדָקָה/צְעָקָה or not. Well, there are two puns in there, hence my confusion. I updated my post. Thank you!
– please delete meJul 9 '14 at 18:30

Most of these are from Chumash. My favorite is in Shmuel Aleph 15:14. Shaul was commanded to wipe out Amalek, including all of their animals. Shmuel came to see how Shaul had done. Shaul greeted Shmuel delighted, implying he had completed his command. Shmuel responds:

And now, I assign to you one portion more than to your brothers, which I wrested from the Amorites with my sword and bow.”

Yaakov tells Yosef he is giving him a "Shechem" over his brothers. Onkelos and Rashi explain this to mean portion, since the number that follows it is masculine, and cities are feminine. However, what is the portion he is getting? Rashi says the city of Shechem!

I'm wondering if אברהם gave 7 sheep to אבימלך as witness that the wells belong to אברהם (Breishis 21:30), and then אברהם and אבימלך swore their treaty, and באר שבע is named because of the swearing (שבועה). I wonder if אברהם picked 7 animals because the word שבע with a סגול sounds like שבע with a פתח, and it's just a play on words.

Also ובן משק ביתי הוא דמשק אליעזר (Breishis 15:2). Depending on how you translate the words בן משק and דמשק, if they aren't related on a pshat level, but are put together because they sound similar, that may be a play on words.

Edit: I thought of another possibility. יצחק is named so because of the laughing of אברהם and שרה, which are different but share the meaning "laughing". Then שרה says "שחוק עשה לי", a different type of laughter. Then ישעמאל is מצחק - something that is also a very different type of laughter.

See R' S. R. Hirsch on the meaning of שבע, and why the word is used for “seven” and “oath”; he relates this homonym to the word אלה, which also means “oath” but is more obviously related to אל, “God”. To Rav Hirsch, your first example is symbolism, not a pun.
– J. C. SalomonFeb 10 '14 at 0:05